Hilbert's Projective Metric and the Norm on a Banach Space

Abstract:

      In this paper, we establish some relations between the Hilbert's projective metric and the norm on a Banach space and show that the metric and the norm induce equivalent convergences at certain set. As applications, we utilize the main results to discuss the eigenvalue problems for a class of positive homogeneous operators of degree $\alpha$ and the positive solutions for a class of nonlinear algebraic system.